Invariant Tori in Hamiltonian Systems with Impacts

Abstract:

It is shown that a large class of solutions in two-degree-of-freedom Hamiltonian systems of billiard type can be described by slowly varying one-degree-of-freedom Hamiltonian systems. Under some non-degeneracy conditions such systems are found to possess a large set of quasiperiodic solutions filling out two dimensional tori, which correspond to caustics in the classical billiard. This provides a unified proof of existence of quasiperiodic solutions in convex billiards and other systems with impacts including classical billiard in electric and magnetic fields, dual billiard, and Fermi–Ulam systems.